Haynsworth inertia additivity formula explained

In mathematics, the Haynsworth inertia additivity formula, discovered by Emilie Virginia Haynsworth (1916–1985), concerns the number of positive, negative, and zero eigenvalues of a Hermitian matrix and of block matrices into which it is partitioned.^[1]

The inertia of a Hermitian matrix H is defined as the ordered triple

In (H) = \left( \pi(H), \nu(H), \delta(H) \right)

whose components are respectively the numbers of positive, negative, and zero eigenvalues of H. Haynsworth considered a partitioned Hermitian matrix

H = \begin{bmatrix} H _{11} & H _{12} \ H ^{\ast} _{12} & H _{22} \end{bmatrix}

where H₁₁ is nonsingular and H₁₂^* is the conjugate transpose of H₁₂. The formula states:^[2]

In \begin{bmatrix} H _{11} & H _{12} \ H ^{\ast} _{12} & H _{22} \end{bmatrix} = In (H _{11} ) + In (H/H _{11} )

where H/H₁₁ is the Schur complement of H₁₁ in H:

H/H _{11} = H _{22} - H ^{\ast} _{12} H ^{-1} _{11} H _{12} .

Generalization

H ^{+} _{11} instead of H ^{-1} _{11} .

The formula does not hold if H₁₁ is singular. However, a generalization has been proven in 1974 by Carlson, Haynsworth and Markham,^[3] to the effect that

\pi(H) \ge \pi(H _{11} ) + \pi(H/H _{11} ) and \nu(H) \ge \nu(H _{11} ) + \nu(H/H _{11} ) .

Carlson, Haynsworth and Markham also gave sufficient and necessary conditions for equality to hold.

See also

• Block matrix pseudoinverse
• Sylvester's law of inertia

Notes and References

1. Haynsworth, E. V., "Determination of the inertia of a partitioned Hermitian matrix", Linear Algebra and its Applications, volume 1 (1968), pages 73–81
2. Book: Zhang, Fuzhen . The Schur Complement and Its Applications . limited . 15. 2005 . Springer. 0-387-24271-6 .
3. D. . Carlson . E. V. . Haynsworth . T. . Markham . A generalization of the Schur complement by means of the Moore–Penrose inverse . SIAM J. Appl. Math. . 16 . 1 . 1974 . 169–175 . 10.1137/0126013 .